Using now a well known Birgé's or Huber's lemma, [1]}, we get
\(\sup _{\omega \in \lbrace 0,1\rbrace ^{m}}\operatorname{P}_{\omega }^{\otimes M}(\widehat{\tau }_{M}\ne \tau ^{\ast })\ge \left[ 0.36\wedge \left( 1-\frac{MK_{\mathcal {H}}}{\log (\left|\mathcal {H}\right|)}\right) \right] ,\) 
